In a saucepan, combine 1 cup of water and 1 cup of sugar. Heat over medium heat, stirring until the sugar is completely dissolved. Remove from heat and let cool.

In a mixing bowl, combine the fresh lemon juice, lemon zest, and 1 cup of water. Stir well.

Once the syrup has cooled, combine it with the lemon mixture. Stir until well mixed. Refrigerate for at least 2 hours until chilled.

Pour the chilled mixture into an ice cream maker and churn according to the manufacturer's instructions until it reaches a soft-serve consistency.

Transfer the sorbet to an airtight container and freeze for at least 2 hours to firm up before serving.